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1、DigitalImageProcessingChapter4:ImageEnhancementintheFrequencyDomain22June2005Background:FourierSeriesAnyperiodicsignalscanbeviewedasweightedsumofsinusoidalsignalswithdifferentfrequenciesFourierseries:FrequencyDomain:viewfrequencyasanindependentvariable(Image
2、sfromRafaelC.GonzalezandRichardE.Wood,DigitalImageProcessing,2ndEdition.FourierTr.andFrequencyDomainTime,spatialDomainSignalsFrequencyDomainSignalsFourierTr.InvFourierTr.1-D,ContinuouscaseFourierTr.:Inv.FourierTr.:FourierTr.andFrequencyDomain(cont.)1-D,Discr
3、etecaseFourierTr.:Inv.FourierTr.:u=0,…,M-1x=0,…,M-1F(u)canbewrittenasorwhereExampleof1-DFourierTransformsNoticethatthelongerthetimedomainsignal,TheshorteritsFouriertransform(ImagesfromRafaelC.GonzalezandRichardE.Wood,DigitalImageProcessing,2ndEdition.Relatio
4、nBetweenDxandDuForasignalf(x)withMpoints,letspatialresolutionDxbespacebetweensamplesinf(x)andletfrequencyresolutionDubespacebetweenfrequenciescomponentsinF(u),wehaveExample:forasignalf(x)withsamplingperiod0.5sec,100point,wewillgetfrequencyresolutionequaltoTh
5、ismeansthatinF(u)wecandistinguish2frequenciesthatareapartby0.02Hertzormore.2-DimensionalDiscreteFourierTransform2-DIDFT2-DDFTu=frequencyinxdirection,u=0,…,M-1v=frequencyinydirection,v=0,…,N-1x=0,…,M-1y=0,…,N-1ForanimageofsizeMxNpixelsF(u,v)canbewrittenasorwh
6、ere2-DimensionalDiscreteFourierTransform(cont.)Forthepurposeofviewing,weusuallydisplayonlytheMagnitudepartofF(u,v)(ImagesfromRafaelC.GonzalezandRichardE.Wood,DigitalImageProcessing,2ndEdition.2-DDFTProperties(ImagesfromRafaelC.GonzalezandRichardE.Wood,Digita
7、lImageProcessing,2ndEdition.2-DDFTProperties(cont.)(ImagesfromRafaelC.GonzalezandRichardE.Wood,DigitalImageProcessing,2ndEdition.2-DDFTProperties(cont.)(ImagesfromRafaelC.GonzalezandRichardE.Wood,DigitalImageProcessing,2ndEdition.2-DDFTProperties(cont.)(Imag
8、esfromRafaelC.GonzalezandRichardE.Wood,DigitalImageProcessing,2ndEdition.ComputationalAdvantageofFFTComparedtoDFTRelationBetweenSpatialandFrequencyResolutionswhereDx=spatialresolutioninxdirectio
